Revolutionaries and Spies II: Hypercubes & Complete Multipartite - PowerPoint PPT Presentation

Revolutionaries and Spies II: Hypercubes & Complete Multipartite Graphs Douglas B. West Department of Mathematics University of Illinois at Urbana-Champaign west@math.uiuc.edu slides available on DBW preprint page Joint work with Jane V. Butterfield, Daniel W. Cranston, Gregory Puleo, and Reza Zamani

A Game of National Security Two teams: r revolutionaries and s spies on a graph G .

A Game of National Security Two teams: r revolutionaries and s spies on a graph G . Start: Each rev and then each spy occupies a vertex.

A Game of National Security Two teams: r revolutionaries and s spies on a graph G . Start: Each rev and then each spy occupies a vertex. Round: Each rev and then each spy moves or doesn’t.

A Game of National Security Two teams: r revolutionaries and s spies on a graph G . Start: Each rev and then each spy occupies a vertex. Round: Each rev and then each spy moves or doesn’t. Goal: Revs. want a meeting of size m unguarded by spies; spies want to prevent this.

A Game of National Security Two teams: r revolutionaries and s spies on a graph G . Start: Each rev and then each spy occupies a vertex. Round: Each rev and then each spy moves or doesn’t. Goal: Revs. want a meeting of size m unguarded by spies; spies want to prevent this. Def. RS ( G, m, r, s ) is the resulting game; who wins? Invented by Beck

A Game of National Security Two teams: r revolutionaries and s spies on a graph G . Start: Each rev and then each spy occupies a vertex. Round: Each rev and then each spy moves or doesn’t. Goal: Revs. want a meeting of size m unguarded by spies; spies want to prevent this. Def. RS ( G, m, r, s ) is the resulting game; who wins? Invented by Beck Obs. s ≥ min { | V ( G ) | , r − m + 1 } ⇒ spies win. Spies can sit on all vertices or follow all but m − 1 revs.

A Game of National Security Two teams: r revolutionaries and s spies on a graph G . Start: Each rev and then each spy occupies a vertex. Round: Each rev and then each spy moves or doesn’t. Goal: Revs. want a meeting of size m unguarded by spies; spies want to prevent this. Def. RS ( G, m, r, s ) is the resulting game; who wins? Invented by Beck Obs. s ≥ min { | V ( G ) | , r − m + 1 } ⇒ spies win. Spies can sit on all vertices or follow all but m − 1 revs. Obs. s < min { | V ( G ) | , ⌊ r/m ⌋ } ⇒ revs win. Revs can make more meetings than spies can guard.

A Game of National Security Two teams: r revolutionaries and s spies on a graph G . Start: Each rev and then each spy occupies a vertex. Round: Each rev and then each spy moves or doesn’t. Goal: Revs. want a meeting of size m unguarded by spies; spies want to prevent this. Def. RS ( G, m, r, s ) is the resulting game; who wins? Invented by Beck Obs. s ≥ min { | V ( G ) | , r − m + 1 } ⇒ spies win. Spies can sit on all vertices or follow all but m − 1 revs. Obs. s < min { | V ( G ) | , ⌊ r/m ⌋ } ⇒ revs win. Revs can make more meetings than spies can guard. Ques. Fix G, m, r . How many spies are needed to win?

Spy-Good Graphs Def. G is spy-good if ⌈ r/m ⌉ spies win, for all r, m .

Spy-Good Graphs Def. G is spy-good if ⌈ r/m ⌉ spies win, for all r, m . • Trees are spy-good. (Proved also by Howard & Smyth)

Spy-Good Graphs Def. G is spy-good if ⌈ r/m ⌉ spies win, for all r, m . • Trees are spy-good. (Proved also by Howard & Smyth) • Unicyclic graphs are spy-good. ⌊ r/m ⌋ spies also win if the one cycle is short enough.

Spy-Good Graphs Def. G is spy-good if ⌈ r/m ⌉ spies win, for all r, m . • Trees are spy-good. (Proved also by Howard & Smyth) • Unicyclic graphs are spy-good. ⌊ r/m ⌋ spies also win if the one cycle is short enough. • Graphs with a dominating vertex  are spy-good. Spies wait at  except when guarding meetings elsewhere.

Spy-Good Graphs Def. G is spy-good if ⌈ r/m ⌉ spies win, for all r, m . • Trees are spy-good. (Proved also by Howard & Smyth) • Unicyclic graphs are spy-good. ⌊ r/m ⌋ spies also win if the one cycle is short enough. • Graphs with a dominating vertex  are spy-good. Spies wait at  except when guarding meetings elsewhere. • Interval graphs are spy-good ( ⌊ r/m ⌋ spies suffice).

Spy-Good Graphs Def. G is spy-good if ⌈ r/m ⌉ spies win, for all r, m . • Trees are spy-good. (Proved also by Howard & Smyth) • Unicyclic graphs are spy-good. ⌊ r/m ⌋ spies also win if the one cycle is short enough. • Graphs with a dominating vertex  are spy-good. Spies wait at  except when guarding meetings elsewhere. • Interval graphs are spy-good ( ⌊ r/m ⌋ spies suffice). • Chordal graphs?

Spy-Bad Graphs Def. G is spy-bad if r − m spies lose, for some r, m .

Spy-Bad Graphs Def. G is spy-bad if r − m spies lose, for some r, m . • For all r, m , some chordal graph is spy-bad. • • • • • m r � r • • • • • • • • • • • • • • • � m

Spy-Bad Graphs Def. G is spy-bad if r − m spies lose, for some r, m . • For all r, m , some chordal graph is spy-bad. • • • • • m r � r • • • • • • • • • • • • • • • � m Revs initially occupy the vertices of the clique.

Spy-Bad Graphs Def. G is spy-bad if r − m spies lose, for some r, m . • For all r, m , some chordal graph is spy-bad. • • • • • m r � r • • • • • • • • • • • • • • • � m Revs initially occupy the vertices of the clique. Spies can’t reach all threatened meetings outside. Some m unguarded revs can meet on the first round.

Spy-Bad Graphs Def. G is spy-bad if r − m spies lose, for some r, m . • For all r, m , some chordal graph is spy-bad. • • • • • m r � r • • • • • • • • • • • • • • • � m Revs initially occupy the vertices of the clique. Spies can’t reach all threatened meetings outside. Some m unguarded revs can meet on the first round. Thought: spy-bad means dense enough and sparse enough for revs to threaten some unreachable mtg.

Random Graphs Thm. For fixed r, m , the random graph is almost surely spy-bad ( r − m spies lose, r − m + 1 spies win).

Random Graphs Thm. For fixed r, m , the random graph is almost surely spy-bad ( r − m spies lose, r − m + 1 spies win). Pf. The revs occupy some r vertices.

Random Graphs Thm. For fixed r, m , the random graph is almost surely spy-bad ( r − m spies lose, r − m + 1 spies win). Pf. The revs occupy some r vertices. The r − m spies occupy some set S , size at most r − m .

Random Graphs Thm. For fixed r, m , the random graph is almost surely spy-bad ( r − m spies lose, r − m + 1 spies win). Pf. The revs occupy some r vertices. The r − m spies occupy some set S , size at most r − m . Some set T of m vertices has unguarded revs.

Random Graphs Thm. For fixed r, m , the random graph is almost surely spy-bad ( r − m spies lose, r − m + 1 spies win). Pf. The revs occupy some r vertices. The r − m spies occupy some set S , size at most r − m . Some set T of m vertices has unguarded revs. In the random graph, almost surely, for every set S of size r − m and every set T of size m , some vertex  is adjacent to all of T and none of S .

Random Graphs Thm. For fixed r, m , the random graph is almost surely spy-bad ( r − m spies lose, r − m + 1 spies win). Pf. The revs occupy some r vertices. The r − m spies occupy some set S , size at most r − m . Some set T of m vertices has unguarded revs. In the random graph, almost surely, for every set S of size r − m and every set T of size m , some vertex  is adjacent to all of T and none of S . The revs meet at  in the first move and win.

Hypercubes Thm. For m = 2 , the hypercube Q d is spy-bad if d > r .

Hypercubes Thm. For m = 2 , the hypercube Q d is spy-bad if d > r . Pf. V ( Q d ) = { 0 , 1 } d . Vertices of weights 1 , 2 , 3 are singles, doubles, triples. Claim r − 2 spies can’t win.

Hypercubes Thm. For m = 2 , the hypercube Q d is spy-bad if d > r . Pf. V ( Q d ) = { 0 , 1 } d . Vertices of weights 1 , 2 , 3 are singles, doubles, triples. Claim r − 2 spies can’t win. � doubles. � r Revs start at r singles, threatening at 2

Hypercubes Thm. For m = 2 , the hypercube Q d is spy-bad if d > r . Pf. V ( Q d ) = { 0 , 1 } d . Vertices of weights 1 , 2 , 3 are singles, doubles, triples. Claim r − 2 spies can’t win. � doubles. � r Revs start at r singles, threatening at 2 r − 2 spies at singles can’t reach all threats at doubles. • • • • • • • 1 r • ∅

Hypercubes Thm. For m = 2 , the hypercube Q d is spy-bad if d > r . Pf. V ( Q d ) = { 0 , 1 } d . Vertices of weights 1 , 2 , 3 are singles, doubles, triples. Claim r − 2 spies can’t win. � doubles. � r Revs start at r singles, threatening at 2 r − 2 spies at singles can’t reach all threats at doubles. • • • • • • • 1 r • ∅ ≤ r − 5 spies at singles leave too many threats at doubles (spies at triples reach only three doubles).